Question

A certain acid, HA, has a vapor density of 5.11 \(\mathrm{g} / \mathrm{L}\) when in the gas phase at a temperature of \(25^{\circ} \mathrm{C}\) and a pressure of 1.00 atm. When 1.50 \(\mathrm{g}\) of this acid is dissolved in enough water to make 100.0 \(\mathrm{mL}\) of solution, the \(\mathrm{pH}\) is found to be \(1.80 .\) Calculate \(K_{\mathrm{a}}\) for the acid.

Short answer

The acid dissociation constant (\(K_a\)) for the given acid is calculated as 0.002456.

Step-by-step solution

Determine the molar mass of the acid HA

To determine the molar mass of the acid, we can use the ideal gas law. The ideal gas law equation is: \[PV = nRT\] where P is the pressure, V is the volume, n is the amount of substance (in moles), R is the ideal gas constant, and T is the temperature in Kelvin. We are given the vapor density of the gas (\(\rho\)), which is the mass per unit volume (\(\rho = \frac{m}{V}\)). We can rearrange the ideal gas law to solve for the molar mass (\(M = \frac{m}{n}\)). \[PV = \frac{m}{M}RT\] Rearrange to solve for molar mass: \[M = \frac{mRT}{PV}\] Molar mass can be calculated as follows: - Convert the temperature from Celsius to Kelvin: \(T = 25^{\circ}C + 273.15 = 298.15\,\text{K}\) - R (ideal gas constant) = 0.0821 L atm mol⁻¹ K⁻¹ - P (pressure) = 1.00 atm - mass (m) = 5.11 g/L × V - volume (V) = 1 L (to simplify the calculation) Plug in values and solve for the molar mass: \[M = \frac{(5.11\,\text{g/L} \times 1\,\text{L}) \times (0.0821\,\text{L}\,\text{atm}\,\text{mol}^{-1}\,\text{K}^{-1}) \times (298.15\,\text{K})}{(1.00\,\text{atm}) \times (1\,\text{L})}\] \[M = 126.97\,\text{g/mol}\]

Determine the concentration of HA

Given that 1.50 g of HA is dissolved in enough water to make 100.0 mL of solution, we can calculate the concentration (in moles per liter) as follows: \[C = \frac{n}{V}\] where C is the concentration, n is the amount of substance in moles, and V is the volume in liters. Since we now have the molar mass of HA, we can convert the mass of HA into moles: \[n = \frac{1.50\,\text{g}}{126.97\,\text{g/mol}} = 0.01182\,\text{mol}\] Now, convert the volume from milliliters to liters: \[V = \frac{100.0\,\text{mL}}{1000\,\text{mL/L}} = 0.100\,\text{L}\] Now we get: \[C_{HA} = \frac{0.01182\,\text{mol}}{0.100\,\text{L}} = 0.1182\,\text{M}\]

Calculate the concentration of H+ ions and A- ions

The pH of the solution is given as 1.80. Using the definition of pH, we can calculate the concentration of H+ ions: \[\text{pH} = -\log{[H^+]}\] Rearrange and solve for [H+]: \[[H^+] = 10^{-(\text{pH})} = 10^{-1.8} = 0.01585\,\text{M}\] Now, we need to calculate the A- ions concentration. Since the dissociation reaction of the acid is given by: \[HA \rightleftharpoons H^+ + A^-\] This means that when HA loses one H+ ion, it forms one A- ion. Therefore, [A-] = [H+] = 0.01585 M.

Calculate Ka

Now we have the initial concentration of HA and the concentrations of H+ and A- ions. The equilibrium equation for the dissociation of the acid is: \[K_a = \frac{[H^+][A^-]}{[HA]}\] Using the calculated concentrations: \[K_a = \frac{(0.01585\,\text{M})(0.01585\,\text{M})}{(0.1182\,\text{M} - 0.01585\,\text{M})} = \frac{0.0002515}{0.10235} = 0.002456\] Thus, the acid dissociation constant (\(K_a\)) for the acid is 0.002456.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and amount of a gas. It is given by the formula \(PV = nRT\).

• \(P\) stands for pressure, typically measured in atmospheres (atm).
• \(V\) is the volume the gas occupies, generally given in liters (L).
• \(n\) represents the number of moles of the gas.
• \(R\) is the ideal gas constant, which is usually 0.0821 L atm mol⁻¹ K⁻¹.
• \(T\) is the temperature in Kelvin.

To use this equation, you often need to rearrange it to find the unknown quantity by isolating each variable as needed. It's crucial to ensure that all units are consistent \- you may need to convert grams to moles, Celsius to Kelvin, or other standardizations like maintaining consistent units of volume and pressure.

pH Calculation

Calculating the pH of a solution allows us to understand its acidity or basicity. pH is a measure of the hydrogen ion concentration in a solution and is calculated using the equation \(\text{pH} = -\log{[H^+]}\).

• If the pH is below 7, the solution is considered acidic.
• A pH above 7 indicates a basic (alkaline) solution.
• A pH of exactly 7 is neutral, like pure water.

In the given problem, calculating the pH involves determining the concentration of hydrogen ions \([H^+]\). This is done by using the formula \([H^+] = 10^{-\text{pH}}\). By knowing the pH, you can work backwards to find the exact concentration of hydrogen ions in the solution.

Molar Mass

Molar mass is the mass of one mole of a substance, often expressed in grams per mole (g/mol). It's vital for converting between mass and moles using the formula:\[n = \frac{m}{M}\]Here, \(n\) is the number of moles, \(m\) is the mass in grams, and \(M\) is the molar mass. For any substance in the gaseous state, the ideal gas law can help determine its molar mass. By rearranging the equation \(PV = \frac{m}{M}RT\), molar mass can be calculated as \(M = \frac{mRT}{PV}\). This method is particularly useful when dealing with gases where volume, pressure, and temperature are known, allowing for the calculation of the molar mass directly from these measurable properties.

Chemical Equilibrium

Chemical equilibrium occurs in a reversible reaction when the rates of the forward and reverse reactions are equal, resulting in constant concentrations of the reactants and products. It is expressed through the equilibrium constant \(K\), which for acids and bases is known as \(K_a\) for acids and \(K_b\) for bases.The formula for \(K_a\) in the context of an acid dissociation reaction \(HA \rightleftharpoons H^+ + A^-\) is: \[K_a = \frac{[H^+][A^-]}{[HA]}\]

• \([H^+]\) and \([A^-]\) are the concentrations of the product ions at equilibrium.
• \([HA]\) is the concentration of the undissociated acid remaining.

In the problem, to find \(K_a\), we need the concentration of ions in solution at equilibrium, analogous to how we calculated for [H+] using pH. Understanding equilibrium constants is crucial because it provides insight into the extent of dissociation for weak acids and bases, helping to predict the behavior of acids in various concentrations.